1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 11 Integer, Goal, and and Nonlinear Programming

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students will be able to Understand the difference between LP and integer programming. Apply the cutting plane method to solve integer programming problems. Understand and solve the three types of integer programming problems Apply the branch and bound method to solve integer programming problems. Solve goal programming problem graphically and using a modified simplex technique. Formulate nonlinear programming problems and solve using Excel.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter Outline 11.1 Introduction 11.2 Integer Programming 11.3 Modeling with 0-1 Variables 11.4 Goal Programming 11.5 Nonlinear Programming

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Linear Programming Extensions Integer Programming Linear, integer solutions Goal Programming Linear, multiple objectives Nonlinear Programming Nonlinear objective and/or constraints

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Fig. 11.1 X2X2 X1X1 6 5 4 3 2 1 0 123456 + + ++++ ++ + possible Integer Solution 6X 1 + 5X 2 <  30 2X 1 + 3X 2  12 Optimal LP Solution (X 1 = 3 3/4, X 2 = 1 1/2 Profit = $35.25)

6. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Three Types of Integer Programming Problems 1.Pure integer programming problems All variables integer. 2.Mixed-integer programming problems Some variables integer. 3.Zero-one integer programming problems All variables either 0 or 1.

7. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Integer Programming Techniques Gomory’s Cutting Plane Method Branch and Bound Method

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Fig. 11.2 X1X1 X2X2 6 5 4 3 2 1 0 123456 + + ++++ ++ + = possible Integer Solution 6X 1 + 5X 2  30 2X 1 + 3X 2  12    XX :Cut Optimal LP solution

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Steps in Branch & Bound 1. Solve problem using LP. If solution is integer - finished. If not - upper bound. 2. Find any feasible integer solution to get lower bound. 3. Branch on noninteger variable from step 1. Split problem into two pieces: integer above, and integer below. 4.Create nodes at top of these branches by solving the new problems.

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Steps in Branch & Bound - Continued 5. a) Branch solution not feasible, terminate branch. b) Branch solution feasible, not integer, go to step 6. c) Branch solution feasible, integer, check. If equal to upper bound - solution. If less than upper bound, but greater than lower bound - new lower bound and proceed. If less than lower bound - terminate branch.

11. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Branch & Bound - Continued 6. Check branches. New upper bound is maximum of objective at all final nodes. If upper bound equals lower bound, stop; if not, go to step 3.

12. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Company First Branching         Subject to: :Max     X XX XX XX Subproblem A          XX XX XX Subject to: :Max Original Problem        Subject to: :Max     X XX XX XX Subproblem B

13. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Company Second Branching         Subject to: :Max     X XX XX XX Subproblem A                X X XX XX XX Subject to: :Max Subproblem C                X X XX XX XX Subject to: :Max Subproblem D

14. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Company Third Branching         X X       X X   X X XX Subject to: :Max Subproblem D    XX                X X X XX XX Subject to: :Max Subproblem E    XX                X X X XX XX Subject to: :Max Subproblem F

15. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Company Branch & Bound - Overall X 1 =3.75 X 2 =1.5 P=35.25 X 1 =4 X 2 =1.2 P=35.20 X 1 =4.16 X 2 =1 P=35.16 X 1 =5 X 2 =0 P=35.00 X 1 =4 X 2 =1 P=34.00 X 1 =3 X 2 =2 P=33.00 A B D C E F No Feasible Solution Original LP Solution

16. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Goal Programming Versus Linear Programming Multiple Goals (instead of one goal) Deviational Variables Minimized (instead of maximizing profit or minimizing cost of LP) “Satisficing” (instead of optimizing) Deviational Variables are Real (and replace slack variables)

17. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Harrison Electric Company Goal Programming d  Subject to:           XX XX XX : :Max Original Problem Subject to:             XX XX ddXX d : :Min Problem with goal of Profit = 30  +

18. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Analysis of the Profit Goal 123456 X2X2 X1X1 5 4 3 2 1 0 Minimize Z = d 1 - - d 1 + d1+d1+ d1-d1-    XX     XX    XX 6 7

19. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Analysis of Priority Goals Minimize Z = P 1 d 1 - + P 2 d 2 - + P 3 d 3 + + P 4 d 4 - X2X2 X1X1 5 4 3 2 1 0 d1+d1+ d1-d1- 123456    XX    XX    XX 6 7 d3+d3+ d3-d3- d2-d2- d2+d2+    X d4+d4+ d4-d4-

20. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Initial Goal Programming Tableau

21. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Second Goal Programming Tableau

22. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Solution to Harrison Electric’s Goal Programming

23. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Nonlinear Programming Nonlinear objective function, linear constraints Nonlinear objective function and nonlinear constraints Linear objective function and nonlinear constraints

24. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Nonlinear Programming Nonlinear objective function, linear constraints           X.X. X X X.XX :toSubject :Max

25. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Nonlinear Programming Nonlinear objective function and nonlinear constraints                  XX XX XX X XXXX :toSubject :Max

26. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Nonlinear Programming Linear objective function and nonlinear constraints Subject to:                  XX. XX X.XX.X XX :Min

27. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 11-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Computational Procedures -Nonlinear Programming Gradient method (Steepest descent) Separable programming - linear representation of nonlinear problem